A binary operation * is defined as the set,R,of a real number by m*n=m+n+2.find the identity element and inverse of n under thr equation 

We are given the binary operation \( * \) defined on the set \( \mathbb{R} \) of real numbers by \( m * n = m + n + 2 \). We need to find both the identity element and the inverse of a number \( n \) under this operation.

### 1. Finding the Identity Element

To find the identity element \( e \in \mathbb{R} \), we require:

\[
m * e = m \quad \text{for all} \quad m \in \mathbb{R}
\]

Substituting the operation into the equation:

\[
m * e = m + e + 2 = m
\]

To solve for \( e \), subtract \( m \) from both sides:

\[
e + 2 = 0
\]

Thus, the identity element is:

\[
e = -2
\]

### 2. Finding the Inverse of \( n \)

The inverse of a number \( n \) under this operation is a number \( n' \in \mathbb{R} \) such that:

\[
n * n' = e
\]

Given that the identity element \( e = -2 \), we substitute into the operation:

\[
n * n' = n + n' + 2 = -2
\]

To find \( n' \), solve the equation:

\[
n + n' + 2 = -2
\]

Subtract 2 from both sides:

\[
n + n' = -4
\]

Now, solve for \( n' \):

\[
n' = -4 - n
\]

So, the inverse of \( n \) under this operation is:

\[
n' = -n - 4
\]

### Summary:
- **Identity Element**: \( -2 \)
- **Inverse of \( n \)**: \( -n - 4 \)

**Would you like more details or have any questions?**

Here are 8 related questions that expand on this topic:

1. How do you verify that \( n' = -n - 4 \) is indeed the inverse of \( n \)?
2. Can you find the inverse of \( n \) if the operation was \( m * n = m + n - 2 \)?
3. How does the identity element affect the structure of the operation in a group?
4. Is this operation associative? How would you prove or disprove it?
5. What would the inverse of \( n = 3 \) be under this operation?
6. Can you express the operation \( m * n = m + n + 2 \) in a different form?
7. What is the result of \( 5 * (-3) \) under this operation?
8. How do properties of this operation compare to standard addition on \( \mathbb{R} \)?

**Tip:** To understand binary operations deeply, always analyze the existence and uniqueness of identity and inverse elements, as these are key properties in the context of group theory.

Learn how to find the identity element and inverse of a number under a binary operation defined on the set of real numbers. This problem involves understanding binary operations, identity elements, and inverse elements in mathematics.

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